NEW Conjecture 1.19 (universal nesting of tire-tread trees,
sketch):
For any two rooted trees of tire treads T_1 = T(G_1, S_1) and
T_2 = T(G_2, S_2), T_1 NESTS into T_2:
Choose any tire T in T_2 and any non-trivial bounded face f of
its inner outerplanar graph O^(T). Then there exists a maximal
planar graph G̃ with level source S̃ such that:
(N1) T(G̃, S̃) contains T_2 as a sub-tree.
(N2) The sub-tree rooted at the new child of T at face f is
isomorphic to T_1.
Informally: any tree of tire treads can be inserted into any
non-trivial face slot of any other tree of tire treads. The
class of trees of tire treads is closed under composition by
face-slot insertion.
Followed by Remark 1.20 motivating the conjecture:
- Compositional colourability: if 4-colourability of G̃ follows
from 4-colourability of G_1, G_2 via parent-child consistency
(Remark 1.18 / former tree-coloring-factorisation), then 4CT
propagates through nesting. A min 4CT counterexample would have
to be irreducible under such nesting.
- Universality: trees of tire treads become a "term algebra" for
decomposing plane triangulations; coloring arguments can be
inductive on this algebra.
Open subquestions in remark:
- Precise notion of "isomorphic as rooted trees of tire treads"
(combinatorial vs geometric vs up to embedding).
- Constructive description of G̃ from G_1, G_2, f.
- Compatibility with Birkhoff's internally 6-connected condition.
Page count: 12 → ~13.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
6413560a7b